Minecraft – What does the “Bonus Chest” option in the New World options do

The world will just generate with air blocks replacing non-vanilla blocks, and your non-vanilla items will just disappear. Re-installing the mod will not bring the blocks or items back, unless you do so before loading the world again.

(As a side note, some mods used to render your world unplayable unless the mod was installed again, I'm pretty sure Notch fixed it though, but there might be a few that mess things up.)

There doesn't seem to be any official way. Then again, modding the game by splaying open and modifying the internals of the .jar file was never really officially supported, either. The new launcher is, ultimately, part of a rather large series of changes to Minecraft, including making mods actually something the game supports via the new Resource Pack stuff, but until that time, modding is a bit more awkward.

However, there's still a way to do it. Go into the versions folder inside .minecraft, and copy the version you want to mod. Rename the copied folder, and both the files inside, to some new name, say "modded 1.blah". Lastly, open the .json file in a text editor, and change the line that will read something like "id": "1.6.1", so that the ID matches the new name. The new launcher should now show a version "modded 1.blah" in the list, but since that version doesn't exist on the Minecraft servers, it won't get overwritten. Splay open and tinker with the .jar file in there the same way you always would with the old launcher.

Note:
If you are using the new launcher with an older version of Minecraft, then the mods you are using may attempt to access files in the bin folder directly. If this is the case, you can simply create a folder named 'bin' in the .minecraft folder where they are looking for it, and add the files the mod needs in there. The mod should find the files, and carry on as it used to, not knowing the difference.